There was a workshop on this problem at Univ. Marburg, in March 2017:
https://www.mathematik.uni-marburg.de/~agricola/Hopf2017/
It resulted in a special issue of the Journal of Differential Geometry and its Applications (April 2018):
https://www.sciencedirect.com/journal/differential-geometry-and-its-applications/vol/57/suppl/C
Most of the papers from that issue are on the ArXiv. The introductory one is a good historical overview:
Ilka Agricola, Giovanni Bazzoni, Oliver Goertsches, Panagiotis Konstantis, Sönke Rollenske, On the history of the Hopf problem, arXiv:1708.01068
The pdf of that paper has arxiv links to most of the other papers (or see below). It mentions the papers by Etesi and Atiyah, saying about them that "the community of experts does not seem to find unity" and linking these MO threads.
By all appearances though, the problem is still open ;-).
The rest of the papers are:
Panagiotis Konstantis, Maurizio Parton, Almost complex structures on spheres, arXiv:1707.03883
Cristina Draper, Notes on $G_2$: The Lie algebra and the Lie group, arXiv:1704.07819
Ilka Agricola, Aleksandra Borówka, Thomas Friedrich, $S^6$ and the geometry of nearly Kähler $6$-manifolds, arXiv:1707.08591
Ana Cristina Ferreira, Non-existence of orthogonal complex structures on the round 6-sphere, ResearchGate (request only)arXiv:1906.02062
Boris Kruglikov, Non-existence of orthogonal complex structures on 6-sphere with a metric close to the round one, arXiv:1708.07297
Daniele Angella, Hodge numbers of a hypothetical complex structure on $S^6$, arXiv:1705.10518
Christian Lehn, Sönke Rollenske, Caren Schinko, The complex geometry of a hypothetical complex structure on $S^6$, ResearchGate (request only)
Aleksy Tralle, Markus Upmeier, Chern's contribution to the Hopf problem: an exposition based on Bryant's paper, arXiv:1708.02904
